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Additional resources for The Complexity of Boolean Functions (Wiley Teubner on Applicable Theory in Computer Science)

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12) According to Fig. 1 G-functions are called triangles and V-functions are called rectangles. In S2 we compute some not too large triangles and rectangles. For some parameter τ to be chosen later we partition {0 n − 1} to blocks of size 2 4 2τ and compute the corresponding triangles and rectangles. These results are used in S3 for the computation of all carry 44 bits cj where j = k 2τ −1 for some k . In S4 we fill the gaps and compute all cj . We have already computed triangles of size 1 , namely uj , and rectangles of size 1 , namely vj .

38) We have used the fact that Ak(n) computes pn before the last step. 38) easily follows from induction. 4 : 0 ≤ k ≤ ⌈log n⌉ The prefix problem is solved by Ak(n) . 40) How can we use the prefix problem for the addition of binary numbers ? We use the subcircuits S1 and S5 of Krapchenko’s adder with size 2n and n − 1 resp. and depth 1 each. S1 computes a coding of the inputs bits. 42) We know that cj = uj ∨ vj cj−1 . 44) This looks like the prefix problem. We have to prove that G = ({A(0 0) A(0 1) A(1 0)} ◦) is a monoid.